Three-loop Correction to the Instanton Density. I. The Quartic Double Well Potential

TL;DR

This paper calculates the three-loop quantum correction to the instanton density in a quartic double-well potential, confirming previous results and highlighting the emergence of irrational contributions at higher orders.

Contribution

It provides the first detailed three-loop calculation of the instanton density for the quartic double-well potential, matching earlier results and analyzing complex Feynman integrals.

Findings

Three-loop correction matches previous high-precision results.

Feynman diagrams at three-loop involve irrational contributions.

Higher-order corrections reveal complex integral structures.

Abstract

This paper deals with quantum fluctuations near the classical instanton configuration. Feynman diagrams in the instanton background are used for the calculation of the tunneling amplitude (the instanton density) in the three-loop order for quartic double-well potential. The result for the three-loop contribution coincides in six significant figures with one given long ago by J.~Zinn-Justin. Unlike the two-loop contribution where all involved Feynman integrals are rational numbers, in the three-loop case Feynman diagrams can contain irrational contributions.